The Geometry of Transmission Zeros in Distance-Based Formations

Abstract

This letter presents a geometric input-output analysis of distance-based formation control, focusing on the phenomenon of steady-state signal blocking between actuator and sensor pairs. We characterize steady-state multivariable transmission zeros, where fully excited rigid-body and deformational modes destructively interfere at the measured output. By analyzing the DC gain transfer matrix of the linearized closed-loop dynamics, we prove that for connected, flexible frameworks, structural transmission zeros are strictly non-generic; the configuration-dependent cross-coupling required to induce them occupies a proper algebraic set of measure zero. However, because extracting actionable sensor-placement rules from these complex algebraic varieties is analytically intractable, we restrict our focus to infinitesimally rigid formations. For these baselines, we prove that the absence of internal flexes forces the zero-transmission condition to collapse into an explicit affine hyperplane defined by the actuator and the global formation geometry, which we term the spatial locus of transmission zeros. Finally, we introduce the global transmission polygon--a convex polytope constructed from the intersection of these loci. This construct provides a direct geometric synthesis rule for robust sensor allocation, guaranteeing full-rank steady-state transmission against arbitrary single-node excitations.

Figures (2)

• Figure 1: Global Transmission Polygon for an asymmetric 4-agent formation. Agent $p_2^*$ lies precisely on $\mathcal{L}_1$, inducing an exact rank loss ($\langle p_1^* - p_{\mathrm{cm}}, p_2^* - p_{\mathrm{cm}} \rangle = -J_p/n = -4$). By reciprocity, $p_1^* \in \mathcal{L}_2$. Agents $p_3$ and $p_4$ reside strictly inside the shaded polygon, guaranteeing omnidirectional signal transmission.
• Figure 2: Singular values of $sG_{ji}(j\omega)$ for the 4-agent formation. The maximum singular values (solid lines) remain strictly positive due to the invariant uniform translational modes. For the infinitesimally rigid framework, the minimum singular value (blue dashed) strictly drops as $\omega \to 0$, confirming a DC rank of 1. Introducing internal flexes breaks this exact geometric cancellation, restoring the full DC rank (red dashed).

Theorems & Definitions (14)

• Proposition 1: Measure-Zero Rarity
• proof
• Definition 1: Spatial Locus of Transmission Zeros
• Theorem 1
• proof
• Remark 1: Directional Signal Blocking
• Remark 2: Contrast with State-Space Uncontrollability
• Corollary 1: Actuator-Sensor Reciprocity
• proof
• Definition 2: Global Transmission Polygon